generalized gamma measures and shot-noise cox processes

Generalized Gamma Measures and Shot-Noise Cox ProcessesAuthor(s): Anders BrixSource: Advances in Applied Probability, Vol. 31, No. 4 (Dec., 1999), pp. 929-953Published by: Applied Probability TrustStable URL: http://www.jstor.org/stable/1428335 .

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Adv. Appl. Prob. (SGSA) 31, 929-953 (1999) Printed in Northern Ireland

? Applied Probability Trust 1999

GENERALIZED GAMMA MEASURES AND SHOT-NOISE COX PROCESSES

ANDERS BRIX,* Royal Veterinary and Agricultural University

Abstract

A parametric family of completely random measures, which includes gamma random measures, positive stable random measures as well as inverse Gaussian measures, is defined. In order to develop models for clustered point patterns with dependencies between points, the family is used in a shot-noise construction as intensity measures for Cox processes. The resulting Cox processes are of Poisson cluster process type and include Poisson processes and ordinary Neyman-Scott processes. We show characteristics of the completely random measures, illustrated by simulations, and derive moment and mixing properties for the shot-noise random measures. Finally statistical inference for shot-noise Cox processes is considered and some results on nearest-neighbour Markov properties are given.

Keywords: Spatial point process; random measure; stable measure; gamma measure; gamma process; shot-noise process; Cox process; Poisson cluster process; Neyman- Scott process AMS 1991 Subject Classification: Primary 60G57

Secondary 60G57; 60G60; 60J30; 65C05

1. Introduction

Clustering point patterns often occur in applications such as botany, where a plant sets seed around itself, resulting in a cluster of plants of this species the following year. This clustering mechanism is the idea behind the construction of Neyman-Scott point processes (Diggle (1983); Stoyan et al. (1995)) where each point in a stationary Poisson process of parent points gives rise to a stochastic number of offspring points, independently distributed around the parent point according to a specified density f. If the number of offspring for each parent point is Poisson distributed, the Neyman-Scott process is also a Cox process, that is a Poisson process driven by a stochastic intensity measure M. In the Neyman-Scott case M is given as M(dx) = X ? f(x - xi) dx, where the xi's are the points of a stationary Poisson process, and X is the expected number of offspring for each parent point. The measure M is an example of a shot-noise measure, and the Neyman-Scott process is the simplest example of a shot-noise Cox process. In general a shot-noise process is a marked point process which has been kernel smoothed using the marks as weights (see for example Rice (1977)). A shot-noise Cox process is a Cox process with shot-noise intensity measure.

Cox processes arise as models for clustered point patterns in another context as well, namely as generalizations of mixed Poisson distributions: if samples from a clustered point pattern are considered, the counts of points in the samples will show overdispersion, i.e. the variance

Received 30 March 1998; revision received 19 October 1998. * Postal address: Guy Carpenter Instrat, Aldgate House, 6th Floor, 33 Aldgate High Street, London, EC3N 1AQ. Email address: [email protected]

929



930 * SGSA A. BRIX

will exceed that of the Poisson distribution. Such overdispersion is often modelled by mixed Poisson distributions, where the mean of the Poisson distribution is taken to be a random variable. In a point process context this corresponds to letting the intensity of a Poisson process be stochastic, which is the definition of a Cox processes.

Shot-noise Cox processes thus arise naturally as models for clustering, both in a biological sense and as extensions of the usual way of dealing with overdispersion.

We describe the G-family of infinitely divisible random measures and consider simulation and inference for Cox processes driven by these random measures. The G-family is defined as completely random measures from a family of infinitely divisible distributions given by Hou- gaard (1986) which generalizes the gamma distributions. The G-family of random measures thus includes gamma random measures, but also random measures with marginal distributions following inverse Gaussian, non-central chi-square or positive stable distributions. Similar constructions have previously been used by for example Samorodnitsky and Taqqu (1994) for stable random measures and Moran (1959) for gamma processes. In fact most results given in this paper generalize to other random measures defined in the same way from other infinitely divisible distributions, since the construction is the equivalent of Levy processes (or additive processes, see L6vy (1954)) for random measures.

The G-random measures have two important properties. They are almost surely discrete and they are completely random, i.e. they have independent increments. When used as intensity measures for Cox processes they thus result in point processes which are again completely random and which consist of multiple points. For many applications one would however want simple point processes with dependencies between points, so in order to achieve this, shot- noise G-measures are defined as kernel smoothed G-measures.

Using shot-noise G-measures as intensity measures for Cox processes we introduce a large family of point process models for clustered point patterns. The family includes both Poisson processes, gamma-Poisson random fields, which have recently been considered by Wolpert and Ickstadt (1998), and ordinary Neyman-Scott processes. It extends gamma-Poisson random fields to, for example, stable and inverse Gaussian-Poisson random fields, and it includes cluster processes of general Neyman-Scott type. The family may also be shown to include certain nearest-neighbour Markov point processes (Baddeley and Moller 1989; Baddeley et al. 1996).

Theoretical properties are easily derived for both G-measures and the corresponding Cox processes, mainly via the Laplace functional and Campbell's theorem. For instance, explicit formulas for all moment measures can be given and stationarity and Markov properties can be shown. Finally shot-noise G Cox processes are easy to simulate and extend easily to multitype and spatio-temporal Cox processes.

Being a generalization of well-known and well used point process models, the family described in this paper is of theoretical interest, but as shown by Aalen (1992), Hougaard (1986) and Hougaard et al. (1997), extensions of existing models using the G-family can be very useful for modelling purposes. These authors demonstrated how the G-family can be used to extend existing overdispersion models for count data and frailty models in survival analysis, and gave examples of how to use the models on medical data. Likewise, the models described here have a variety of potential uses for modelling physical and biological spatial phenomena: for example, Brix and Chadceuf (1998) use a spatio-temporal extension of shot-noise G Cox processes, for modelling the positions of weeds in a field.

Apart from extending many known models, the family has the advantage of being very flexible. It allows non-stationarity to be modelled in a natural way, for example through the



G-measures and shot-noise Cox processes SGSA * 931

non-random shape measure (see Section 2), or through the shot-noise kernel, which may be non-stationary, allowing spatially varying scale parameters on the intensity of the Cox processes.

The paper is organized as follows. First the G-family of generalized gamma measures is defined followed by the extension to shot-noise G-measures. Expressions for the moments of shot-noise G-measures are given and a result on stationarity and mixing is given. In Section 4 Cox processes driven by G-measures are discussed. Simple G-measures resulting in point processes with multiple points are briefly discussed, then Cox processes driven by shot-noise G-measure are studied. Examples and simulations are provided and Markov properties and methods for statistical inference are discussed. Eventually the results are discussed in Sec- tion 5.

2. The generalized gamma measures

The random measures in this section are based on a family of probability distributions on [0, xc) suggested independently by Tweedie (1984) and Hougaard (1986), and also considered by Bar-Lev and Enis (1986), Jorgensen (1987) and Aalen (1992). The family can briefly be characterized as the natural (or tilted) exponential family generated by the positive stable distributions, or as the positive valued subclass of the exponential family with variances which are power functions of the means. The two articles by Hougaard (1986) and Aalen (1992) give thorough treatments of the family. Reviews of the properties of the family (and of the P-G-family, see Section 4) can be found in Hougaard et al. (1997) and Brix (1997).

The Laplace transform of the family is

L,,s,o(s) = exp

(-((0

+ S)a - a0), s > 0, (1)

where a < 1, 8 > 0 and 0 > 0. For a < 0 the family is defined only for 0 > 0 and Lo,8a, is defined to be the limit of La,a,o as a --* 0, i.e. Lo,8a, = (0/(0 + s))6, which is the Laplace transform of the gamma distribution with scale parameter 0-1 and shape 8. Letting a < 0 gives a compound Poisson distribution (Aalen (1992)) which is a convolution of a Poisson distributed number of gamma random variables with scale parameter 0-1 and shape -a, while 0 < a < 1 corresponds to the natural exponential family generated by the positive stable distributions (the stable distributions arise when 0 = 0). Finally a = 1 gives the distribution which is degenerate in 3, and thus does not depend on the parameter 0. We will denote the distribution with Laplace transform (1) by G(a, 3, 0), and the whole family will be called the G(ca, 8, 0)- family. The members of the G((a, 8, 0)-family are infinitely divisible in the parameter 3 and concentrated on [0, 0 ), which makes them a natural basis for a family of random measures with independent increments.

2.1. G-measures

Consider a Borel subset E c Rd equipped with its Borel o-algebra 8. We shall say that a measure is boundedly finite if it is finite on bounded sets.

Definition 2.1 (G-measures) Let (a, 0) e (0, 1] x [0, oc) U (-~o, 0] x (0, oc) and let c be a non-negative, boundedly finite measure on (E, 8). A random measure, /, on (E, G) is said to be a G-measure with index a, shape measure K and intensity parameter 8 (or for short a G(a, K, 8)-measure) if the following conditions hold:



932 * SGSA A. BRIX

(a) (b)

(0"

00d(0

0 0

o 0

0 0 C0/s "4. 1 '

oI I

:s ~ ~ ~ ?

............

r .... :s

- •,?

,t. o. ,? o. "k , " "%

0. "?.- .0.

.5.

'0 S '06 ~3

FIGURE 1: Simulations of G(a, K, 0)-measures with 0 = 0.1 and K the Lebesgue measure and

(a)a• = 0.5, (b) a = 0, (c) a = -0.5 and (d) a = -2.

(i) t (A) follows a G(a, K(A), 0) distribution for every bounded A E 8.

(ii) it has independent increments, that is, it(A1), . g.., (Ak) are mutually independent whenever A I...., Ak are disjoint.

Using Daley and Vere-Jones (1988) Theorem 6.1.VI, for example, it is seen that G-measures actually exist. By the infinite divisibility of the G-family and the independent increment property (ii), one can construct a consistent family, (1t(A))AEG, of random variables such that it(Ai U A2) = A(A1) + i(A2) a.s., for disjoint and bounded A1, A2 e &. The continuity of K

ensures that iL(An) -- 0 a.s. for any sequence (An) of bounded Borel sets, with A 4 0, since the G-family is continuous in distribution (see Hougaard (1986), Lemma 2).

Note that G-measures are infinitely divisible, and that 0 actually is the inverse scale parameter when a = 0, i.e. when i is a gamma measure.

Example 2.1 (Hougaard processes and gamma measures) Lee and Whitmore (1993) define a Hougaard process on the line as a stochastic process (Xt)t>o with independent G(a, s, 0) distributed increments Xt+s - Xt. A Hougaard process is thus a G((a, K, 9)-measure on [0, 00oo) with K taken to be the Lebesgue measure.

The case a = 0 corresponds to X being an ordinary gamma process or more generally a stationary gamma measure on Rd, while a = 0.5 corresponds to an inverse Gaussian measure.

For 0 = 0.1 and K the Lebesgue measure and for four different values of a (a = -2, -0.5, 0, 0.5), Figure 1 shows realizations of G(a, K, 0)-measures on [-0.5, 1.5] x [-0.5, 1.5].




o(a) (b)

o 0O

co*

s o~

•~: ?n .i :'•:,.., . . :::;;.

,• .2''. ....:-:

.. .... ..

,t, o "':I] '. ::"

(

"?2e .o.b

FIGURE 2: Simulations of G-measures with a = 0, K the Lebesgue measure and (a) 0 = 0.01, (b) 0 = 0.1 and (c) 0 = 1.

Needles indicate positions of atoms with the masses of atoms marked by the height of the needles. The realizations will be used later in examples of shot-noise G-measures on [0, 1] x [0, 1], which is why this subset of the plane has been marked by a quadrat. For a = 0 and a = 0.5 the realizations contain infinitely many atoms (by Corollary 2.1) and only the 750 atoms with the largest masses are shown. Note the effect of changing a. For a = 0 a few large atoms are present and many very small; increasing a to 0.5 'homogenizes' the sizes of the atoms. Letting a decrease to -0.5, the smallest atoms disappear and the measure only has a finite number of atoms. Further decreasing a to -2 gives rise to more and larger atoms.

Keeping a = 0 fixed (a stationary gamma measure), Figure 2 shows realizations of G(a, K, 9)-measures with 0 = 0.01, 0.1 and 1 (K still being the Lebesgue measure). It is seen how 0 acts as an inverse scale parameter, since all three plots show the same structure regarding the proportion between the sizes of the atoms, but on three different scales.

The role of the measure K is that of a spatial intensity, and it can for example be used to introduce a deterministic trend in the model, the idea being that more of the large atoms will appear in regions with large K-measure.

When a = 1 we get the non-stochastic G(1, K, 9)-measure i which equals the measure K. In the more interesting case a < 1, the distribution of i can be characterized by its Laplace transform. The L6vy representation of the Laplace transform, La,K(A),o, for the G(ca, K(A), 0)- distribution is given by

log La,K(A),(s) = f (exp(-sy) - 1)vA(dy), (2)



934 * SGSA A. BRIX

where the Levy measure vA is the measure with density

[ y-(-) exp(-Oy)

with respect to the Lebesgue measure on [0, oo). The L6vy representation (2) is useful for deriving theoretical properties of G-measures, as

the following theorem shows.

Theorem 2.1 (Properties of G-measures) Let it be a G(a, K, 0)-measure with a < 1. Then the following hold for it:

(i) IX

can be represented in the form /t(A) = fyo yN(A x dy), where N is a Poisson measure on E x [0, oo) with intensity measure v given by

v(A x B)= K(A) y-'-1 0y dy, A e , B e S([0, 00)). (3)

(ii) it is almost surely purely atomic.

(iii) tI has no fixed atoms if and only if K is diffuse (a random measure it is said to have a fixed atom at x if P (tIt(x) > 0) > 0).

(iv) If E = Rd then /tt is stationary if and only if K is proportional to the Lebesgue measure on E.

Proof Let N be a Poisson measure on E x [0, 0o) with intensity measure v given by (3) and consider the measure/2 on E given by

A(A) = yN(A x dy), A E 8. (4)

The Laplace transform, LA, of jA(A) is

LA(s) = Eexp(- fo sylA(x)N(dx, dy))

= exp (j (e-sy - 1)A(dy) , (5)

and it is seen that AA follows the same distribution as gI(A). The Laplace functional LA of ti can be found by first calculating it for simple functions by means of (2) and then taking limits. This gives

L,(f) = exp (e-f(x)y - 1)v(dx, dy) (6)

for all bounded and measurable functions f on E. Since 4(A) and 2(A) have the same Laplace transform it is thus seen that they have the same Laplace functional; they consequently follow the same distribution. The properties (ii) and (iii) now follow from the representation Theorem 6.3.VIII in Daley and Vere-Jones (1988).

Now let E = Rd, let K be the Lebesgue measure and define fu (x) = f(x + u) for all x, u e Rd . Simple calculations show that because of the translation invariance of the Lebesgue



G-measures and shot-noise Cox processes SGSA e 935

measure we have that Lg(f) = Lg(fu) for all u e Rd, consequently it is stationary by Daley and Vere-Jones (1988), Theorem 10.1.III. Conversely, since the Lebesgue measure is the only translation invariant measure on Rd, the same theorem shows that in order for pI to be stationary, K must be the Lebesgue measure.

The representation (i) in Theorem 2.1 is useful for understanding G-measures since it says that we can write a G(a, K, 0)-measure pI as

W -

i xi (7)

where Sx is the Dirac measure at x, and where the xi follow the distribution of the atoms of a Poisson measure with intensity measure proportional to K. The masses (wi) of the atoms can be obtained as the jump times for an inhomogeneous Poisson process on the line, and the following corollary will show that the sum (7) consists of infinitely many terms when 0 < a < 1. When a < 0 the number of terms in (7) is finite and v(A x dy) is proportional to the density of a gamma distribution with shape parameter -a and scale parameter 1/0. It follows that each atom mass wi follows a gamma distribution with shape -a and scale 1/0.

We use the representation (7) later, for constructing shot-noise G-measures and for deriving a method for simulating G-measures.

Corollary 2.1 (Support of a G-measure) Let IL be a G(a, K, 0)-measure with a < 1 and define Ne (A) to be the number of atoms for gi in A E 8 with mass exceeding E > 0:

NE(A) := I{x E A : L(x) > }1l,

and denote by S, (A) the support of It within A.

N,(.) is a Poisson process on (E, 8), with finite intensity measure v,(.) := v(. x [e, oo)), and S. (A) is dense in the support of K when a e [0, 1), while the number of points in S. (A) is finitefor every bounded set A and a < 0.

Proof With the notation from Proposition 2.1, Ne can be written as Ne (A) = N(A x [E, 00)) showing that Ne is a Poisson process with intensity v,. Defining N as the almost sure limit N = lim,,o N8, we get a random measure which is not necessarily boundedly finite. However

F(a) _K(A) lim v (A) = K(A)- - < x00 E+O '(1 - a) -a

when a < 0, and thus N is a Poisson process with finite intensity measure vo. When a e [0, 1) we have that v, (A) -- +oo as s O0, so that N(S,L (A)) is almost surely unbounded for every Borel set A.

2.2. Shot-noise G-measures

G-measures are interesting in their own right as models for discontinuous random fields or considered as marked point process models (where the marks are the atom masses), but here the issue of modelling point processes and count data is addressed, and we will use the G-measures for constructing intensities and intensity measures for Cox processes. As men- tioned in Theorem 2.1, G-measures are completely random, purely atomic measures, and when used as intensity measures for Cox processes this results in completely random point processes with multiple points. While desirable for some modelling purposes this is not the



936 * SGSA A. BRIX

case when modelling e.g. positions of trees or weeds. Instead in the following we will study kernel smoothed G-measures whereby diffuse measures are obtained, and disjoint regions may be stochastically dependent.

To obtain a general definition of what we will call a shot-noise G-measure, we use the representation (i) in Theorem 2.1 to identify a G(ca, K, 0)-measure It with a Poisson process N on E x [0, 00) with intensity v given by (3).

Definition 2.2 (Shot-noise G-measures) Let E, S be two Borel subsets of d and let 8 and 4 be the corresponding Borel a-algebras. Consider a G(a, IK, 0)-measure, A, on E with a < 1. Let D : (4, Ex [0, oo)) --+ [0, x ) be a kernel from Ex [0, x ) to S satisfying fE x[0,0) D(A, u) x v(du) < 0 for all bounded A E 8. Write It as

/t = Ei wi•xi and define the boundedly

finite measure M on S by

M(A) = - D(A, (xi, wi)), A -.

M is called a shot-noise G-measure with control measure It. For a = 1, the corresponding shot-noise G-measure is the non-stochastic measure M given by M(A) = fE D(A, x) dK (x).

An important class of shot-noise G-measures are those which have kernels of the form D (A, (x, w)) = fA (s, x)w ds, since for these measures it is possible to calculate moment measures explicitly. We shall call these measures standard shot-noise G-measures.

Note that for a standard shot-noise G measure, the parameter 0 is identifiable if and only if a 0, since, for a = 0, we obtain the same model as defined by Definition 2.2 if we take 0 = 1 and use the kernel D(., (x, w))/0. Furthermore it is possible to let the kernel D be stochastic, but we shall not pursue this approach any further here.

Example 2.2 (Linear processes) Let S C E = -ld and let m be a measure on E x [0, 09), g a measurable function on E x [0, 00) and assume that D (-, (x, w)) has the density of the form g(. - x, w) with respect to m. The measure M has a density which is a linear process:

M(ds) = g(s - xi, wi) ds. (8)

If, for example, g(-, w) is a continuous function on f, then the shot-noise measure M can be considered as a smoothing of the Lt-measure, as can be seen from Figure 3.

A very appealing property of shot-noise measures of the form (8) is that they are known to be stationary and ergodic, provided that A is stationary (Proposition 2.2), a property which is inherited by a Cox processes with M as intensity measure.

Example 2.3 (Randomized Lebesgue measure) Another way to construct a stationary shot- noise measure is by considering a kernel which is proportional to the Lebesgue measure (here denoted by ?), i.e. a(-, (x, w)) = we(-). In this case we also have to require that K(E) < co in order to make M almost surely boundedly finite. Then M(A) = ?(A) y wi, and it is easily seen that M becomes stationary, because of the translation invariance of the Lebesgue measure, but it is not ergodic.

Recall the simulations of the four G-measures on [-0.5, 1.5] x [-0.5, 1.5] in Figure 1.




(a) (b)

tO C-'"-" "-..

N o " . . ... . ? ....... .. . .

. . .. . . . ...

o 0

o o

NOO

a) N

u3 oJ

(c)

............ ...(d) ..

O 0O

0 0? ?

o 0.. . ....

(C ~ ....?-?~ o~~??.. ~ ~ o

- ...

0

-- o? i-

r

0,1 _gL

0.~

FIGURE 3: The shot-noise construction (2.2) applied to each G-measure from Figure 1, with D the isotropic Gaussian kernel with density (9). y = 50 and p is chosen so that 0 a-1p = 100.

Figure 3 shows the density of the shot-noise G-measure obtained by applying the shot-noise construction (2.2) to each of these realizations. In all four cases a Gaussian kernel with density s - 4(s, (x, w)) has been used, where

d(s, (x, w)) = PY

e-Yllx-sl2w, (9) 7r

with y = 50 and p chosen so that 0a-lp = 100 (if the G-measures were defined on IR2, then Oa-1p would be the mean of M([0, 1] x [0, 1])). By defining the shot-noise G-measure on [0, 1] x [0, 1] and using y = 50 and G-measures defined on [-0.5, 1.5] x [-0.5, 1.5], border effects become negligible and the realizations shown in Figure 3 can be considered as driven by G-measures defined on R2.

Proposition 2.1 (Moments of shot-noise G-measures) For 0 > 0, the kth order cumulant measure, XkM, of a shot-noise G-measure M is given by

Mk1 )Eo

k XkM(A1 ...Ak)

=F1 -(Ai, (x, w))w-a-1 e-ow

dw K(dx). i=1

In particular,

we have thatfor a standard shot-noise G-measure, with ~(., (x, w)) = (., x) w, the intensity and covariance measures are given by



938 * SGSA A. BRIX

EM(A) = 0"-I'

f (A, x)K(dx),

cov(M(Ai), M(A2)) = 0a-2(1 - a)

fE

(Ai, x) (A2, x)K(dx).

When 0 = 0 and 0 < a < 1 only the fractional moments of order k < a exist.

Proof Let X~N denote the kth cumulant measure, then since N is completely random a Campbell theorem shows that

k

xkM(A1

....

Ak) =Ef rE I)I H (Ai, ui)zN(du) ....N(duk) ~x [0,) ki=1

k

]E-O- H (Ai, ui)Xk(dul,.., duk) =

Ex[0, )k i=1k(u1

. .9

i=1

since a Poisson measure has all cumulant measures equal to its intensity measure. When 0 < a < 1 and 0 = 0, the measure t is a stable measure with index a in which case only the fractional moments of order less than ao exist.

As noted in Example 2.3, the randomized Lebesgue measure is non-ergodic since one event M(A) completely determines another, M(B), as soon as ?(A) and ?(B) are known. On the other hand the measures in Example 2.2 are always ergodic, as can be seen from the following proposition.

Proposition 2.2 (Mixing properties) Assume that S = E = Id, and let M be a shot-noise G-measure with kernel D and stationary control measure il. If D as a function from E to S is a convolution kernel, then M is stationary and mixing.

Proof Let cie be the random measure consisting of those atoms (xi, wi) of ~ for which wi - A(xi) > E, and let Me be the random measure obtained from (2.2) by replacing N

by N(,. n [E, oo)). Denote by LM, and L,,, the Laplace functionals of M and i respectively. Since ,e is completely random (and thereby mixing) it follows from Daley and Vere- Jones (1988), Proposition 10.3.VI, that for two measurable, non-negative functions, fi, f2, of bounded support

LAE (fi + Tv f2) -+ L ,L (fl)LL (f2), as IlvJ -- oO,

where Tv is the shift operator (T, f)(x) = f(x + v). Defining lf : (x, w) ? fs f (s)D(ds, (x, w)), it is easily seen that the Laplace functional

LM, takes the form LM, (f) = L,, ((Iff), and that (for f, f2 as above)

LM(fl + T f2) =

Eexp(-Jf (tyfI(u) ?

Tfz(u)) I(du)). Ex[O, I)

If c is a convolution kernel, it holds that

4iT((x, w)) = Jf2(s + v)O(ds, (x, w)) =f f2(s)(ds, (x + v, w)) = Tohf((x, w)),




and thus

LM, (fl + Tvf2) = Lit,(/fi + Tvo/f2) ----- L8(lrfi)LL8, (f2) = LM,(fl)L, (f2),

as Ilvll -- o. Appealing once more to Daley and Vere-Jones (1988), Proposition 10.3.VI shows that M is

mixing, while replacing fi by the constant function zero shows that M is stationary, since then

LM (Tv f2) = Lit ,(Tv Yf2) = Lt, (*Y2)

= LM, (f2).

The proposition follows by letting E -- 0.

Note that nowhere in the proof is it required that the 'mixing-measure' is a G-measure, moreover the result holds for any 'mixing-measure' provided it is mixing. Each shot-noise G- measure with ( a convolution kernel is thus stationary, but the class of all such measures does not contain all stationary shot-noise G-measures since for example the randomized Lebesgue measures in Example 2.3 cannot be written in this form. The question is whether or not there exist stationary shot-noise G-measures other than the randomized Lebesgue measures and those with linear process densities (Example 2.2)-I have not been able to find any.

3. Simulation of G- and shot-noise G-measures

We consider simulation of a shot-noise G(a, K, 0)-measure with control measure Lt on a bounded set A. First we consider simple simulation of a G(a, K, 0)-measure as a completely random measure on a fine grid. Afterwards we consider direct simulation of the atoms of i. From both types of simulation it is possible to obtain simulations of shot-noise G-measures by kernel smoothing the simulated G-measures.

3.1. Simulating G-measures on a grid Since a G-measure is completely random, we can simulate it as independent G(a, 6i, 0)

random variables on a fine grid, where 6i is the K-measure of each cell in the grid. By kernel smoothing the grid of G(a, 6i, 0) random variables we obtain a simulation of a shot-noise G-measure that is approximately valid for standard shot-noise G-measures, for which the kernel (x, w) + D(., x, w) is continuous in x and is linear in w. If the kernel is not linear in w, direct simulation of the atoms of the G-measure should be considered instead, before kernel smoothing (see Section 3.2).

Simulation of G(a, 6, 0) random variables can be done in various ways. Independent random variables from a G(a, 6, 0)-distribution can be simulated in several ways; Bondesson (1982) and Daimen et al. (1995) use approximate methods valid for all ao < 1, but simulation from the exact distributions is possible if different strategies are employed depending on the value of a.

For a = 0 the G(ao,, 0)-distribution is a gamma distribution for which plenty of simulation methods exist (Ripley (1987)). For a < 0 the G(a, 6, 0)-distribution is a compound Poisson distribution (Aalen (1992)) and it can be generated as the convolution of a Poisson distributed number of gamma random variables with scale parameter 1/0 and shape parameter -a. Finally for a > 0 the G(a, 6, 0)-family is the exponential family generated from the positive stable distributions, so that rejection sampling (Ripley (1987)) can be applied. Generate stable random variables Yi with Laplace transforms L(t) = exp(-6ta/a) and accept them with probability exp(-0 Yi); the accepted random variables will then follow a G (a, 6, 0)- distribution. Chambers et al. (1976) give a fast method for simulating stable random variables.



940 * SGSA A. BRIX

3.2. Simulating the atoms of a G-measure

Let il be a G(a, K, 0)-measure on some subset E of Id. By Theorem 2.1 it can be represented as a Poisson measure on xd x [0, oo) with intensity measure v given by

v(A x B) = K(A) F o

y-a e-0y dy. (10)

If a < 0 the integral in (10) is finite for all bounded Borel sets A C E and B C [0, 00). It follows from (10) that the atoms of it are distributed in E according to a Poisson process on E with intensity measure KOa/-a, and that the atom masses are independent and identically gamma distributed with shape parameter -oa and scale parameter 0-1. If 0 < a < 1 the integral in (10) is infinite for subsets B of the form B = [0, E] for all E > 0, which by Corollary 2.1 means that the discrete measure iL has dense support on the support of K. Exact simulation of G(a, K, 0)-measures for 0 < a < 1 is thus impossible, and we shall in the following section show how approximate simulation can be done.

3.3. Simulating a G-measure for 0 < a < 1

In order to simulate it on a bounded subset A C E, we define a measure g on [0, 00) by g(t) = g([t, oo)) = v(A x [t, 0x)), where v is the measure given in (10). Since g is a diffuse measure satisfying g((8, oo)) < +oo, for all e > 0, it is seen that if (Xi, X2,... ) is a sequence of random variables on A with density proportional to K restricted to A, and if (Ti, r2,....) is a standard Poisson process on [0, oo) (i.e. with unit intensity measure), then ((X1, g- (rl)), (X2, g-1(T2)), ... ) has the same distribution as it regarded as a Poisson measure on Id x [0,

o•). Assume from now on that i < r2 ..

', then simulation of it

can be done by simulating the sequence (X1, X2, ...), the Poisson process (rl, r2,...), and transforming the ri by g-i, since then

00

i=1

where = means 'equal in distribution'. Because t F g(t) is a decreasing function it follows that g-' (rt) > g-' (r2) > ..., i.e.

the atoms of it will be simulated in decreasing order. Approximative simulation of Lt is thus a question of determining the number R of atoms to include in the simulation of X. This should be done by evaluating the tail sum

L gi (i), (11) i=R+1

which is the sum of all atom masses that have not been included in the approximation. Note that since no explicit expression for g-1 is available, it will usually be tabulated. This can be done in two ways: g is evaluated at fixed points vi > ... > vk and g-1 is tabulated as g-1(tl) = vl ...,g-' (tk) =

vk, so that either g- (t) - 0 for t > tk or g- is approximated by some explicitly given function for t > tk (see Appendix 5). Here the second possibility should be preferred since then R will be a fixed number rather than a stopping time (as is the case if g-l(t) is set to zero for t > tk).

In Appendix 5 an approximation of g- and an evaluation (in probability) of the tail sum (11) is given; similar evaluations should be used to evaluate the tail sum for the corresponding shot-noise G-measure; in this case the sum to be evaluated is




i=R+1

which depends on the kernel D.

4. Cox processes driven by G-measures

In this section we consider two types of Cox processes driven by G-measures: Cox processes with intensity measures given by G-measures which inherit complete independence from the G-measures, and processes with intensity measures which are shot-noise G-measures and as such may be continuous and allow dependencies between points.

4.1. Cox processes independent of G-measures

Let ~ be a G(a, K, 0)-measure on E and consider a Cox process N with intensity measure it. A special case of this type of process is studied in Lee and Whitmore (1993) and Hougaard et al. (1997) where the intensity measure defined on E = [0, oo) is a stationary G-measure, i.e. K is proportional to the Lebesgue measure. In that context the Cox process can be regarded as a subordinated process, where the observed process is X (T (t)) and X is a Poisson process and T is a randomized time or operational time, which in this case is a Hougaard process (see Lee and Whitmore (1993) for further interpretation and references).

Being purely atomic i makes the Cox process N allow multiple points, which makes it a natural model in applications such as, for example, counts of eggs in spatially observed birds' nests. But the process might as well be used as an approximation of a simple point process if the observations are the number of points in small areas rather than point observations. Indeed, if observations are samples from different areas of E, the complete independence property of it is inherited by N and thus results in models from the P-G-family (Hougaard (1986)).

Considered as a completely random point process, N can be interpreted as a marked Poisson process as follows.

Proposition 4.1. Let X = {x E : N({x}) > 0} denote the point process consisting of the points of N counted without multiplicity and let Mx be the mass of N at x e E. Then X is a Poisson process with intensity measure K ()((0 + 1)- a)/aot for a A 0 and K (.)(log(0 + 1) -

log(0)) for a = 0. The marks, Mx, are independent and identically distributed with probability function p given by

p(m) = a(0 + 1)a-m(m - oa)/(((0 + 1), - 0")F(1 - a)m!) for a 0,

and

p(m) = (0 + 1)-m/((log(0 + 1) - log(0))m) for a = 0.

Proof This is a straightforward generalization of Hougaard et al. (1997), Theorem 2.

4.2. Shot-noise G Cox processes The idea of using shot-noise measures as intensity measures for Cox processes was probably

first explored mathematically by Le Cam (1961) who considered a spatio-temporal model for precipitation, a model which has later been widely used in this area (see the review in Larsen (1996)). Vere-Jones and Davies (1966) used the shot-noise Cox construction on the line to model earthquake occurrences in time; there it is supposed that unobserved epochs



942 * SGSA A. BRIX

(a) (b)

. .. .

o

?o Ir •

0.0 0.4 0.8 0.0 0.4 0.8

o .?

Q. *

O *oo0

?

o *@ 0 @0c *

O 0 0 0 O ?

, *n

O .o

0.0 0.4 0.8 0.0 0.4 0.8

X X

o 00

0t 00) * * 0?

0 0

x x

FIGURE 4: Simulations of shot-noise G Cox processes.

occurring according to a Poisson process trigger events with sizes which are independent random variables and which decrease monotonically according to some smoothing kernel. Recently the idea has been taken up again by Wolpert and Ickstadt (1988) where gamma Poisson random fields are studied in a Bayesian setting for modelling positions of trees in a forest. Finally the shot-noise construction has similarities to the models based on random boolean functions, used for example in stereology (Goulard et al. (1994)).

Note that the term shot-noise is used here in wide sense, i.e. the measures are not necessarily stationary and the smoothing kernel may depend on the atoms of the G-measures. If, however, convolution kernels are used, we get constructions similar to Neyman-Scott processes interpreted as Cox processes.

Here we will give results concerning likelihood functions, Markov properties and inference for shot-noise G Cox processes. We consider only the case a < 1 since shot-noise G (1, K, 0)- measures are non-stochastic and thus the shot-noise G Cox processes actually become Poisson processes with intensity measure c.

To show what kind of spatial point processes it is possible to obtain, Figure 4 shows simulations of four shot-noise G Cox processes. The plots are produced by using the shot-noise G-intensities shown in Figure 3 as intensities for four simulated Poisson processes. Recall




that the shot-noise G intensities in Figure 3 can be considered as intensities for shot-noise G-measures with mean 0a-lp = 100 on [0, 1] x [0, 1], so that the expected number of points for each simulation is 100.

4.2.1. Likelihood functions and inference for G-shot-noise Cox processes. Let M be the G-shot-noise measure driven by a G(a, K, 0)-measure, A, on E C Rd, with kernel ( from E x [0, oo) to some bounded subset S of Rd and a < 1. We allow both K and D to depend on unknown parameters from a parameter space E C IRP, and assume that K is absolutely continuous with density k, while ( (., (x, w)) is assumed to be absolutely continuous with density 4(., (x, w)) for almost all (x, w) E E x [0, c•).

For simplicity it is furthermore assumed that K(E) < 00.

Consider a Cox process Y = (Yl ..., yn) on S with intensity measure M, and denote the distribution of (x, w) = ((xl, wl), (x2, w2),...) by Pxw. Then Y has the following density with respect to a homogeneous Poisson process on S with unit intensity (according to Daley and Vere-Jones (1988), p. 498):

q(y; 4) = JeM(S) H m(yi)Pxw(dx, dw), (12) i=1

where m(s) = =1 4(s, (xi, wi)) is the density of M. The case a E [0, 1). When 0 < a < 1 the measure A has infinitely many atoms, and we cannot calculate the integral (12). Instead we fix R E N and approximate A by gR, the measure given by

R

LR = xi,wi ,

i=1

and M by R

M(A) = (D (A, (xi, wi)). i=1

Here xR = (xl,...., xR) and wR = (wl,..., wR) are independent, the xi's are independent

and identically distributed with density k/K(E) and wi = g-'(ri), i = 1,..., R, with g and (ri) defined in Section 3.3. The distribution of R can be found fairly easily as follows:

The jump time ri is gamma distributed, so if T1, T2, ... is a sequence of independent exponential distributed stochastic variables with mean 1, the distribution of ri follows that of T1 + - -

... + Ti. Hence the distribution of wR can be found by transforming T1 ... TR with the transformation f : RR > *RR given by

f(tl, ....

tR) = (g-l(tl)

.... g-1(ti + + tR)).

It follows that if we put lR = {(wl ..., R) : Wl < 2 " < W R}, then the distribution

of (xR, wR) has density

R

pR(X, w; 10) = (F(1 - a))-R e-g(wR) H(k(xi)w--i e-wi)1ER(X)1IR(w)

i=1

with respect to the Lebesgue measure on ER x [0, 00)R



944 e SGSA A. BRIX

The density PR does not converge to anything reasonable as R -+ oc, so instead we consider the following approximation to the likelihood function q:

k

qR(Y)= JERX[0,o)R eMR(S)

mR(yi)PR(xR' wR) dwR dxR,

JERx [0, 00)WR

where mR is the density of MR(.) = ER=1 (, (xi, i)). Using the Laplace functionals for A and AR it is seen that AR converges weakly to A

as R -* ~c (Daley and Vere-Jones (1988), Proposition 9.1.VII). If furthermore (x, w) +

4(s, (x, w)) is continuous for all sES, then the function which maps a discrete measure " on E x [0, oo) into ef D(S,(x,w)) (dx,dw)

-=1 f i 0(yi, (x, w)) (dx, dw), is bounded and continuous, and it follows that qR (y) converges to q (y) as R -- 0*0. The density qR is thus a reasonable approximation to q for large R under this additional assumption.

The case a < 0. When a < 0 the intensity measure, v, for A is finite on E x [0, oo), hence we conclude that the number R of atoms in E is Poisson distributed with probability function

(K(E)Oa/ - a)R r(R; /)

= exp(K(E)0a/-a) R!

It follows from (10) that the atoms (x, w) = ((x1, Wl),..., (xR, R)) have a continuous distribution with density pg given by

p(x, w; F) =

( 0E) F(k(xi)wia( eowi)1ER(X)11R(W).

The likelihood function is thus essentially the same as in the case 0 < a < 1, but with the difference that R is stochastic:

q(y; ') -=

r(R; /fR) )R eM(E) Hm(yi)p(x, w; *) dw dx.

R= JER x[0, oo)R R=0 i=1

Since the likelihood function for both a < 0 and 0 < a < 1 includes a very high dimensional integral, inference based on the likelihood function is not directly feasible. Instead we suggest estimating parameters by the minimum contrast method as proposed in Stoyan and Stoyan (1994) and Moller et al. (1998). In concrete examples the pair correlation function (usually called the g-function, but in order to distinguish it from the function g defined in Section 3.3 we call it gp instead) and the K-function can often both be calculated analytically or at least numerically. Consider for example the isotropic Gaussian kernel given by the density (9), then by Proposition 2.1 the second-order product density p[2] is given by

p[2](s, t) = EM(ds)M(dt)/(ds dt)

= Oa-21 - p2y e-y/21|st2 +(-1 -p)2

Hence (because of stationarity) the pair correlation function gp depends only on the distance between s and t and we can write

g(lls - tl (s, t) -= 1 +-Y0-a - /211s-tl12 h2 T




where X = EM(S)/ISI = - -lp denotes the intensity of the measure M. From this the

K-function is calculated as

K(r) = 2rfsg(s) ds (13)

= 7rr2 +20-a(1 - a) ry(r(o,1(Is) -

(0,1(is) ds),

where we, with a slight abuse of notation, have used oD, 1 to denote the cumulative distribution function for the normal distribution with mean 0 and variance 1.

It turns out that minimum contrast parameter estimation based on the K-function (13) can be numerically unstable and give parameter estimates on the border of the parameter space when there is not enough data to determine the K-function sufficiently well. We have not experi- enced this when estimation is based on the pair correlation function, in fact the minimization algorithm converged very fast in this case. We therefore propose to base parameter estimation on the pair correlation function, which furthermore has the advantage that deviations from the estimated pair correlation function are more easily detected visually than deviations from the estimated K-function, just as it is easier to recognize densities than cumulative distribution functions.

The parameter p does not appear in the expression for K, but since the random measure M is ergodic (by Proposition 2.2) p can be estimated from the empirical density, X, which then converges to the intensity, X, of the point process. The estimate of p becomes

" = -•1-X

a where & and 0 are estimates of a and 0 obtained by the minimum contrast estimation.

4.2.2. Markov property of shot-noise G Cox processes. Besides the theoretical importance of Markov point processes, the Markov property is important for interpretation of models in applications. For example a Markov model can be used to express interactions or competition among individuals in a model for plants or trees. In simple Markov models (Ripley and Kelly (1977)), a neighbourhood is defined to be all points within a fixed radius R (the corresponding relation is called the fixed distance relation), and the Markov property expresses that given the points in its neighbourhood, a point x is independent of the rest of the point process, i.e. R can be considered as an interaction range for the individuals.

In nearest-neighbour Markov models (Baddeley and Mollor (1989); Baddeley et al. (1996)) the neighbourhood is allowed to depend on the configuration of points, and one can for example define the neighbourhood with respect to the connected component relation at distance R, where two points x and y are neighbours if there is a path z1, . Z. , Zk from x to y such that dist(y, zl) < R, dist(zk, x) < R and dist(zi, zi+1) < R. If for example the pattern of positions of weeds in a field is divided into connected components (the connected component for a point x is the collection of all points which are related to this point, see below), then each connected component can be considered as a plant community in which, given the other members of the community, the distribution of each member does not depend on the rest of the weeds in the field.

Recall that when a < 0, a G(a, Kc, 0)-measure g, has only a finite number of atoms on bounded sets, which are all independent. Consider then a shot-noise G-measure M, with control measure g and a convolution kernel t with bounded support. Because of the independence properties of g, a Cox process N with intensity measure M will be a Poisson cluster process. Using a modification of the proof of Theorem 1 in Baddeley et al. (1996), it can be seen that N is actually a nearest-neighbour Markov point process with respect to the connected component relation. This result is stated in Theorem 4.1 below, where it is furthermore shown



946 e SGSA A. BRIX

that in fact the same holds when 0 < a < 1. This is not trivial, since the likelihood is more complicated because there are infinitely many atoms. In order to prove the result we consider the usual setup for spatial Markov point processes, namely the exponential space of our state space S (Carter and Prenter (1972); Baddeley and Mellor (1989); Baddeley et al. (1996)). In this context a realization of a point process is a finite set x = {xl,... x,n} of points, n > 0, for which xi E S, and where multiple points are allowed. A realization is often called a configuration, and the space e of all configurations is the exponential space of S.

Consider a bounded state space S C RI2 and an absolutely continuous kernel D. It follows from Section 4.2.1 that the distribution of a Cox process, y, driven by a shot-noise G intensity measure with kernel D is absolutely continuous with respect to a homogeneous Poisson process on S. This can be expressed as the distribution of y having a density f with respect to the measure ? on C given by

00

'(F) =

- I(yf1QY1, -.Yk}EF)Q(dy1) ...

"(dyk). k=O

Here X is a Borel measure on S. Define for u, v E S the relation - by u - v if dist(u, v) < r. If u -~ v, we say that u and

v are r-close. With this relation we define the connected component relation, ---y, for points yi and yj in a configuration y E C by yi j-y Yj if yi x1l ... xk -- yj for some

xl,..., xk E y. The following definition is due to Baddeley and Mollor (1989).

Definition 4.1 (Baddeley and MEller) A point process y is a nearest-neighbour Markov

process with respect to the connected component relation -"y at distance r if its density f is hereditary (i.e. f(x) > 0 for all x 5 y if f(y) > 0) and for any y E C such that f(y) > 0 and u E S the ratio f(y U u)/f(y) depends only on u, on the y-neighbourhood of u, that is {yi E y U u : u "yu{u} Yi}, and on the relations --y and ~yu{u} restricted to the

y-neighbourhood of u.

The nearest-neighbour Markov properties for shot-noise G Cox processes rely on the following slightly more general result:

Theorem 4.1. Let x = {xl,... x,n} be a Poisson process on EC •Rd

with finite intensity measure X, and let D be a kernel from E to S C Rd such that the support of the measure

SD(., x) is contained in a ball with radius R and centre x for almost all x E E. Assume

furthermore that D(. , x) is absolutely continuous for almost all x E E and consider a Cox process y with intensity measure M given by

n

M(A)= = wi (A, xi),

i=1

where w i, w2, ... is a sequence of independent and identically distributed positive stochastic variables which are independent of x.

Then y is a nearest-neighbour Markov point process with respect to the connected component relation at distance 2R.




Proof The probability generating functional for y is

G(h) = Eexp ( - wi (1 - h(s))O(ds, xi))

Sexp (

fE

[ (L( (1- h(s))(ds, x)) - 1x(dx))

where L w is the Laplace transform of wl. It follows that y is actually a Poisson cluster process with intensity measure X for the parent process and offspring around a parent at a point x distributed according to the density D (., x). The number of offspring points from a parent at x follows a mixed Poisson distribution with probability generating functional

G(s I x) = Lw((1 - s)4a(S, x)).

Theorem 1 in Baddeley et al. (1996) states that a Neyman-Scott process with offspring distribution with support contained in a disk with radius R is a nearest-neighbour Markov point process with respect to the connected component relation at distance 2R. Because of the independence of the wi's, the proof of that theorem generalizes to the case where the offspring distribution at a point xi depends on wi. This means that y is a nearest-neighbour Markov point process with respect to the connected component relation at distance 2R with density

K

f (y) = e- J(YDi), (14)

i=1

where S = fs(1 - q4((0))X(d4), k

k>l ZC1 ....ZCk

j=1

D1,..., DK are the connected components of y, zc, ..., ZCk range over all unordered parti- tions of z into non-empty subconfigurations, and where q? is the density for an offspring point process at a point ?.

Note that since D is assumed absolutely continuous, the densities q? are densities with respect to a standard Poisson measure, fulfilling the requirements for Theorem 1 in Baddeley et al. (1996).

Corollary 4.1. Let y be a shot-noise G Cox process with G(a, K, 0) control measure it and kernel D. If D is absolutely continuous with support contained in a disk with radius R, and ( can be written as D(. , (x, w)) = wc)(. , x), where c is a kernel from E to S, then y is a nearest-neighbour Markov point process with respect to the connected component relation at distance 2R.

Proof For a < 0 Theorem 4.1 can be used directly since the support of g is a Poisson process with finite intensity measure K.

For 0 < a < 1 the situation is a bit more complicated, since the support of g does not constitute a finite Poisson process but rather an infinite Poisson process. Consider instead first the measure ,e consisting of all atoms of

/t which are larger than e > 0:

Wi>s



948 * SGSA A. BRIX

Then the shot-noise Cox process with intensity measure

M() = wiY?(, xi) (xi, wi )E tL,

wi e

clearly satisfies Theorem 4.1. The density fE of the this shot-noise Cox process can of course be written in a way similar

to (12):

fE(Z) = 1[E,)(w) eM(s)fni=

m(zi)Pxw (dx, dw),

and by monotone covergence f, is seen to converge to the density of y as E -- 0. Under the limit, the density f, preserves the structure given in (14) and Lemma 1 in Baddeley et al. (1996) gives that y is a nearest-neighbour Markov point process with respect to the connected component relation at distance 2R.

4.2.3. Multivariate shot-noise G Cox processes. Shot-noise G Cox processes extend easily to the multivariate case by an extension of the definition of shot-noise G-measures. As usual multivariate can mean both multitype point processes, for instance for several species of trees, and spatial point processes developing in time; the definition below covers both cases. There is no problem in extending the measures defined in Section 2 to more general spaces, for example complete separable metric spaces (CSM spaces) (Daley and Vere-Jones (1988)), or locally compact second countable Hausdorff spaces as in Kallenberg (1983). We can thus define a shot-noise G-measure as in Definition 2.2, with the modification that E and S should be some CSM spaces. To construct a multivariate shot-noise G Cox process, we consider as state space the space S = So x I, where So C jRd and I is an index space, and we let E have the same product space structure as S, i.e. E = Eo x I, and E0 c Rd. Letting I = {1,..., k}, the shot-noise measure M will be a random measure which can be used as intensity measure for a multitype Cox process. Consider for example the case k = 2 which corresponds to having two independent G-measures on Eo. The kernel D can be written as D = (I), (2) and the intensity measure for each type is

2 Ni Mi(A) = , Ci(A, (xjk, j, jk)),

j=1 k

where the second sum is over the atoms for the G-measure j. It is seen that for example two independent Cox processes can then be obtained by requiring that (i(', (xjk, j, wjk)) = 0 for j # i. All the correlation structure is thus modelled through the kernel and both positive and negative correlations may be obtained. For example, the covariance measure for the bivariate Cox process Y with intensity measure M and c of the form

ci(', (x, w)) = D)i(-, x)w is

cov(Yi(Ai), Yj(A2))

= 0-2(1 -)( i(A,

x)i).(A2,

)Kl(dx)

where K1, K2 denote the marginal shape measures. Similarly I = [0, oc) would be the natural choice of index space for a spatio-temporal Cox process, in which case the G-measure corresponds to a continuum of independent G-measures.




For both multitype and spatio-temporal processes, the cross covariance measure can easily be calculated using the same type of calculations as in Proposition 2.1 and simulation techniques, results on moment measures, mixing and Markov properties carry over as well with obvious changes.

5. Discussion

Shot-noise G Cox processes provide a flexible family of models for clustering point patterns with a simple and easily interpretable structure. They include Poisson processes and PG-models for count data (Hougaard et al. (1997)) as well as classical Neyman-Scott processes, and in fact the shot-noise Cox construction results in a class of point processes which are general Poisson cluster processes. As such the family provides models which may be interpreted both as germ-grain models (Stoyan et al. (1995)) and as overdispersion models for point data.

Log Gaussian Cox processes (Moller et al. (1998)) represent another flexible and tractable class of models for clustered point patterns. They are uniquely determined by their intensity and pair correlation function, and the pair correlation function may be very similar to some of those obtained by shot-noise Cox processes. This is illustrated in Moller et al. (1998) for the Thomas process, but they find that in general log Gaussian Cox processes and Thomas processes behave very differently from for instance nearest-neighbour functions and empty space functions. Likewise one would expect that other shot-noise G Cox processes have nearest- neighbour and empty space functions which are different from those for log Gaussian Cox processes.

Log Gaussian Cox processes have another appealing property, namely that the intensity can be predicted relatively simply by an empirical Bayes approach using the Metropolis-adjusted Langevin algorithm. The work of Wolpert and Ickstadt (1988) shows that, for a = 0, the intensity of shot-noise G Cox processes can also be predicted by an empirical Bayes approach. Wolpert and Ickstadt (1988) describe gamma-Poisson random fields, which in our terminology means shot-noise G Cox processes with a = 0. Their algorithm for computing posterior distributions is based on a data augmentation algorithm which is not easily extended to other values of a, the problem being that it relies heavily on the distribution of the gamma measure being a conjugate prior. It may be possible however to find other data augmentation schemes in order to make (for example) general G-measures conjugate.

The likelihood function for a shot-noise G Cox process is in general intractable, but since it may be regarded as a missing data problem where the atoms of the G-measure are unobserved, there might be some hope that likelihood inference by Markov chain Monte Carlo methods would be feasible. If so one should be able to simulate realizations from the underlying G-measure given the observed point pattern, which in principle can be done by the Metropolis- Hastings algorithm if it is possible to find a good proposal distribution.

Appendix A. Approximation of g1 and evaluation of a tail sum

This appendix contains two lemmas which will be used to give an evaluation of the tail sum (11). Using the identity g-'(u) = E1 (F(1 - a)/K(A)O-au)/O, the first lemma gives an approximation of the function g-1.



950 * SGSA A. BRIX

Define for 0 <a < 1 the strictly decreasing function E,?+1 : R1 -~1R by

Eat+l (x) = f y-O- e-y dy.

Lemma A.1. Let 0 < x < 1 and u > Ea+1 (1). For 0 < a < 1 it holds that

+l (x) < ox 1( - 1- Ea+1(x) '

-xa 1-a xa

(ua)-l/ 1 - u(1- E (u) < (uaK)-1/'

uo(1 -a))

)a+l For a = 0 we have

x - log(x) - 1 < El (x) < 1 - log(x),

e-1-u < E (u) el-u

Let Fx denote the cumulative distribution function for the gamma distribution with scale 1 and shape parameter

X. > 0, i.e. Fx(x) =

y(X., x)/ F(X), where y(X, x) is the incomplete

gamma function

y(X,x) = fyX-1e-Ydy, > 0.

Lemma A.2. The following inequalities hold for Fx and its inverse F-1' respectively

xx Fx (x) <

F(X + 1)

Fx(u) > (uF(X? + 1))1/.

Theorem A.1. Let 0 < a < 1, 0 > 0 and let (Tk)k>l be the jump times for a homogeneous Poisson process on R1 with unit intensity. Define for M E N

00

RM = g (k),T k=M+1

where g- : R+ - + is given by

g-(u) = 0-1 E-((1

- a)Oau.

For 0 < e < 1 it holds that

P(RM < KM) >1 --s,

where ( 2e a / a

Ml-1/a 0< a< 1

KM- 2 e2

exp[-eM/(2e)] a = 0. 8e




Proof Let e > 0 be given, and let ck denote the E2M-k quantile, k = M + 1, M + 2..., for the gamma distribution with both mean and variance equal to k.

By the Bonferroni bound and the gamma distribution of rk we have that

o00

P(rk < ck for all k > M) < P(rk _<

k) k=M+1

= e2M-k k=M+1

and thus, since g-' is a decreasing function

0

> 1 -8. k=M+1 k=M+1

The theorem is proven if we can bound EVM+1 g-1 (Ck) by KM. (i) Consider the case 0 < a < 1. By Lemma A. 1

E t= +I Ea+(F(1 0) - )0_'Ck) k=M+1 k=M+1

0o

_<-10-1/?t (F(1 - toOk-1/0t

k=M+1

= -1/ar(1 _ )-1/l C•c-1/. (A.1) k=M+1

Since k

~Ilogn k log k - k, n=1

it follows that k!1/k > e-1k and hence by Lemma A.2 that

ck E -e- lk. (A.2) 2

The sum (A. 1) can then be evaluated as

k=M+1 k=M+1

< 2e1/a? _ 1M1/ - 1 oa



952 * SGSA A. BRIX

TABLE 1: M is the number of atoms of the G-measure to include in the simulation. K1 is the bound on the tail sum obtained by Theorem A.1, while K2 is the corresponding bound obtained by direct calculation of the

ck's. The bounds are calculated to hold with probability 1 - e = 0.99.

M 500 1000 1500

K1 752 643 376 322 250 881 M K2 0.021 0.011 0.008

(ii) Now let a = 0. As before we use Lemma A. 1 and (A.2), and we get that

g-l(ck) = 0-1 El1(ck)

k=M+1 k=M+1

<0-1e E exp(-ck) k=M+1

00

<0-1e exp(-e(2e)-lk) k=M+1

2e2 < ~- exp(-EM/(2e)).

The bounds given in Theorem A.1 are very conservative and not of much use in practice. The quantiles ck can however be calculated fairly easily (e.g. in S-Plus), so that a direct calculation of the sum (A. 1) can be made instead of evaluating the ck's by (A.2). This gives a much less conservative evaluation. For a = 0.5, e = 0.01 and four different values of the number M of atoms of the G-measure to include in the simulation, Table 1 shows the bounds on the tail sum (11) obtained by Theorem A. 1, and by direct calculation of the ck's.

Acknowledgements

This paper is a part of the author's Ph.D. thesis, and was supported by a scholarship from The Royal Veterinary and Agricultural University in Copenhagen. Part of the work was done while I was visiting Laboratoire de Biom6trie, INRA, Avignon, France.

I thank Denis Allard, Joel Chadoeuf, Jens Lund, Jesper MOller, Mats Rudemo, Rachid Senoussi, Ib M. Skovgaard and two anonymous referees for helpful comments and corrections.

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generalized gamma measures and shot-noise cox processes

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